Some remarks on the connectivity of Julia sets for 2 - dimensional

We explore the connected/disconnected dichotomy for the Julia set of polynomial automorphisms of C. We develop several aspects of the question, which was first studied by Bedford-Smillie [BS6, BS7]. We introduce a new sufficient condition for the connectivity of the Julia set, that carries over for certain Hénon-like and birational maps. We study the structure of disconnected Julia sets and the associated invariant currents. This provides a simple approach to some results of Bedford-Smillie, as well as some new corollaries –the connectedness locus is closed, construction of external rays in the general case, etc. We also prove the following theorem: a hyperbolic polynomial diffeomorphism of C with connected Julia set must have attracting or repelling orbits. This is an analogue of a well known result in one dimensional dynamics.